Mathematical Sciences: Microlocal Character Theory for Representations of Classical Lie Groups

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Przebinda will investigate distribution characters and matrix coefficients of irreducible unitary representation of classical Lie groups from the viewpoint of microlocal analysis in the context of Howe's theory of reductive dual pairs, via the Cayley transform. Use will be made of Howe's theory, microlocal analysis, and Harish-Chandra's theory of orbital integrals to construct irreducible unitary representations of classical Lie groups, attached to nilpotent coadjoint orbits. This attachment occurs on three levels: associated varieties, wave front sets, and character formulas. The theory of Lie groups, named in honor of the Norwegian mathematician Sophus Lie, has been one of the major themes in twentieth century mathematics. As the mathematical vehicle for exploiting the symmetries inherent in a system, the representation theory of Lie groups has had a profound impact upon mathematics itself, particularly in analysis and number theory, and upon theoretical physics, especially quantum mechanics and elementary particle physics. **//